Calculating the Figure of Merit

Introduction

I have posted a video showing the process involved in calculating the Figure of Merit HERE.

In a previous blog post I discussed the String Test (here) as a means of judging accuracy in the Nineteenth Century.  In this post we will consider a method of judging precision:  The Figure of Merit.  The British 1870 Musketry Manual calls the Figure of Merit a “measure of the efficiency” of shooting (p. 75), and the 1879 Musketry Manual says it is used to conduct experiments with small arms (p. 314).  Thus, it is not a means of judging shooter accuracy (as is the String Test), but rather the precision of the rifle and cartridges, whereas accuracy is a matter of using the results thus obtained for sight adjustment.  The Figure of Merit system was developed in Britain as rifled muskets changed war forever by making excellent accuracy possible over very long ranges, which led to a need to be able to compare different ammunition loads and different weapons to one another.  The Figure of Merit was later adopted in the U.S. as well; Captain Stanhope Blunt discusses its use for U.S. soldiers in his Firing Regulations from 1889.  

"Part XI--Experiments with Small Arms" of the British 1879 Musketry Manual (p. 314), says that: "The quality of arms, projectiles, and gunpowder are now demonstrated by a figure showing the degree of concentration of a certain number of rounds fired which is called by any of the following terms, 'mean deviation,' 'mean absolute deviation,' or 'mean radial deviation.'"  Interestingly, the Manual says the shots should be taken from a rested position, something not done when judging shooter accuracy alone.  (Note that it does not actually use the term "figure of merit" for this system, however, it is the same process described by the Enfield and Whitworth Report discussed below, and both descriptions use the term "mean radial distance.")  The process calculates the mean radial distance of shots from the center of the shot group and thus measures the consistency of a group rather than its accuracy.  It is more useful than just determining the group size because the Figure of Merit averages the distance of the hits from one another, whereas the group size merely indicates the extreme width of the group.  Thus, two groups might be of the same size, but in one the hits might generally be very close together with one or two flyers, whereas in another the hits might be generally farther apart, and the Figure of Merit takes this into account in a way that looking at the group size alone does not.

Definitions

These terms must be understood in order to follow this discussion:

Point of Aim (PoA):  The point at which you aim your sights; this will not necessarily be where you intend your shots to hit since you may have to “hold off” in order to hit a target for which your sights are not set (e.g., you aim low to hit a target slightly closer than the sight setting, or slightly off to one side to allow for a heavy wind).
Mean Point of Impact (MPI):  The average of the group where your shots actually hit; group center.
Intended Mean Point of Impact (IMPI):  Where you intended your group to center.
Group:  The extreme measure of an entire group of shots.

Procedure

The following procedure for calculating the FoM comes from Report of the Enfield and Whitworth Committee published in April of 1861:
“39. It is necessary here to point out that the following plan has been adopted to determine what is called the figure of merit, or the mean radial distance of shots from centre of group.
40. The horizontal distance of each shot upon the target, from a fixed vertical base, is first found, and a mean horizontal distance obtained.
41. The same process is followed to obtain a mean vertical distance from a fixed horizontal base. The intersection of two lines drawn parallel to, and at distances equal to the horizontal and vertical mean distances from the bases, give what is termed the point of mean impact.
42. The distance of each shot upon the target from this point is noted, and, in the case of those recorded as misses, the distance is taken as equal to one-half the diagonal of the target.
43. The whole of these distances, determined as above, are added together, and divided by the number of shots fired, and the result is termed the figure of merit, or mean radial distance of shots from centre of group.”

(See:  Minshall, David.  "Measuring Accuracy."  Research Press. N.D., http://www.researchpress.co.uk/index.php/firearms/british-military-longarms/small-arms-trials/measuring-accuracy)

For the purposes of calculating the Figure of Merit, consider the following hypothetical target:

Using that hypothetical target as an example, here are the steps necessary in order to calculate the Figure of Merit for this ten-shot string (this procedure is taken from the 1879 Musketry Manual pp. 314-316):
1.  Number each hit on the target so that you can track them carefully.
2.  Measure the distance from the center of each hit to the left edge of the target.  In this case, hit #1 is 2.0 inches from the left edge of the target.  Repeat this for every hit and record each measurement in column X in the table below.
3.  Measure the distance from the center of each hit to the bottom edge of the target.  In this case, hit #1 is 8.0 inches from the bottom edge of the target.  Repeat this for every hit and record each measurement in column Y in the table below.

4.  Misses count as 1/2 of the diagonal of the target.  Since the diagonal of this target is 13.9 inches, a miss would count as 6.2 inches for both X and Y, however, in this hypothetical case, there were no misses.
5.  Average all of the X measurements and record the result.
6.  Average all of the Y measurements and record the result.
7.  These averages give you the Mean Point of Impact.  In the case of the example, the average is 3.55 inches for X and 6.83 inches for Y.  Mark the MPI on your target; in this case the point will be 3.55 inches from the left edge of the target and 6.83 inches from the bottom edge of the target. 
8.  Measure the distance from the center of each hit to the MPI and record each measurement in column Z in the table below.   For example, in this case, the distance from hit #1 to the MPI is 1.95 inches.
9.  Finally, average the Z measurements to calculate the FoM; in this case, it is 1.21.  

Shot	X	Y	Z
1	2.00	8.00	1.95
2	2.25	7.25	1.37
3	3.50	7.00	0.18
4	4.00	8.00	1.26
5	2.75	6.50	0.86
6	4.75	6.50	1.24
7	4.50	7.50	1.17
8	3.00	6.00	0.99
9	3.50	5.75	1.08
10	5.25	5.75	2.01
Avg.	3.55	6.83	1.21

Obviously, this system for determining the Figure of Merit is extremely complicated and cumbersome.  Fortunately, Rob Enfield of the British Muzzleloader YouTube channel has done all the hard work of calculating the Figure of Merit by creating a Microsoft Excel worksheet which does it for you.  The spreadsheet and instructions for its use can be found here.

This spreadsheet is both simple to use and extremely powerful.  It eliminates the need to mark the MPI on your target and to have to measure from the hits to the MPI—the worksheet does all of that for you.  All you have to do is to measure the X’s and Y’s, record them on the sheet, and it does the rest.  In addition, if you enter the other information the spreadsheet calls for it will also calculate the sight adjustments necessary to bring your MPI to your IMPI; this is called “zeroing” your rifle.  The spreadsheet also creates a graph showing all of your hits on a diagram which also includes the Point of Aim, Mean Point of Impact, and Intended Mean Point of Impact.  Here is an example of the graph for the hypothetical target used in this essay using Mr. Enfield’s spreadsheet:

Mr. Enfield posted a video explaining the history of the Figure of Merit and giving detailed instructions for the use of his spreadsheet here.

I have a blog post showing how to use the Figure of Merit to work up a load for a specific piece HERE.

Conclusion

The Figure of Merit is an extremely effective method for analyzing the precision of a rifle and its ammunition because it determines the mean deviation of hits from the group center very exactly.  It has two flaws, however:  First, it is extremely tedious and cumbersome, requiring detailed measurements, records, and calculations.  Second, the Figure of Merit only shows the relationship between the shots and the overall group—it does nothing to show the shooter's accuracy, or how close the MPI is to the IMPI.  When shooting, having all your hits close together is excellent, but ultimately, they should be both close together and close to the point you want them to hit; it does little good to have all your shots group tightly together if you miss the target with all of them.  In fairness, Mr. Enfield’s spreadsheet addresses that to some extent by providing sight adjustment information that can be used to bring your MPI to your IMPI, but that is not, strictly speaking, part of the Figure of Merit.  The String Test measurement mentioned above does both, showing not just the size of your group but also how close that group is to your intended target or IMPI, and it does so using a system which is faster and which requires no detailed records or calculations.  Ultimately, both systems are powerful tools which are superior to merely determining group size, and both are historically accurate.